#72321706

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₈
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×C₈×D₄
prime radical	#60673	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #33418 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(3,10)(4,7)(8,13)(9,11)(12,15)(17,18,20,25,22,26,32,39,21,24,28,35,33,38,42,46)(19,27,30,40,31,41,44,48,29,23,34,36,43,37,45,47);(1,2,7,13,5,8,4,6)(3,11,15,14,12,9,10,16)(17,19,28,34,22,31,42,45,21,29,20,30,33,43,32,44)(18,23,35,40,26,37,46,48,24,27,25,36,38,41,39,47);(2,8)(6,13)(9,11)(14,16)(19,29)(23,27)(30,34)(31,43)(36,40)(37,41)(44,45)(47,48);(2,8)(6,13)(9,11)(14,16);(1,3,4,10,5,12,7,15)(2,9,6,14,8,11,13,16)(18,24)(23,27)(25,35)(26,38)(36,40)(37,41)(39,46)(47,48)

Polycyclic presentation:

2^10|a^2=f,b^2=fgi,e^2=gj,f^2=h,g^2=i,h^2=j,b^a=bg,c^b=cij,d^b=di,e^a=eg,e^b=ei,g^a=gi

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321716, #72322147, #72322228
maximal normal subgroup of	#72321716, #72322147, #72322228

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.6*F.7*F.9);;
gap> SetPower(coll, 5, F.7*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);